# Simplify a combination

• Aleoa
In summary, the conversation discusses the simplification of the combination \binom{n}{\frac{n}{2}} and the use of Stirling's formula to characterize its behavior as n becomes larger. The original formula is expressed as 2^{n}(1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})... and the goal is to simplify it. The conversation also mentions the importance of familiarity with Stirling's formula in probability.

#### Aleoa

Member has been warned not to delete the template.
I'm trying to simplify the combination defined as : $\binom{n}{\frac{n}{2}}$.

I did some calculations, starting from the factorial formula $\frac{n!}{(\frac{n}{2})!(\frac{n}{2})!}$ and i found this form :

$2^{n}(1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})...$

but i don't know how to continue, can you help me ?

Aleoa said:
but i don't know how to continue, can you help me ?
No, since you haven't said what your goal is. To me ##\binom{n}{\frac{n}{2}}## is already fine.

I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

Aleoa said:
I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support
In this case I'd try where Stirling's approximation would get me.

Aleoa said:
I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

As fresh_42 suggested, use Stirling's formula. Every student of probability should be thoroughly familiar with that formula, as it is used everywhere.